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  <title>Bang-Bang control</title>
                                                
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<div align="left"><b><u>Description</u></b><br>
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    A basic bang-bang control about single axis was implemented in this scenario. 
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     The model of a linearized spacecraft dynamics with a proportional and
 derivative<br>
     control was considered, which is shown in the illustration below.<br>
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     <img src="BangBang.GIF" alt="Block Diagram" width="584"
 height="233">
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     <b><u>Inputs</u></b><br>
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     <b>J</b>                    &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; Spacecraft Moment 
of inertia<br>
     <b>wn</b>                  &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160; &#160; &#160; Desired natural 
frequency<br>
     <b>damping</b>           &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160; &#160; &#160; Desired damping coefficient<br>
     <b>Torque</b>            &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160; &#160; &#160; &#160; &#160; Torque level of the bang-bang 
actuator<br>
     <b>Dead Zone</b>       &#160;&#160;&#160; &#160;&#160;&#160; &#160; &#160; &#160; Range in which no torque should 
be applied<br>
     <b>theta_com</b>        &#160;&#160;&#160; &#160;&#160;&#160; &#160; &#160; &#160; &#160;Commanded angular position<br>
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     <b><u>Initial Condition<br>
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     <b>theta </b>&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;              &#160; &#160;Spacecraft angular position 
for an actuator  with  no constraint<br>
     <b>thetaDot</b>          &#160;&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160; &#160;Spacecraft angular velocity
for an actuator with   no constraint<br>
     <b>thetaBang </b>&#160;&#160; &#160;&#160;&#160; &#160;&#160;&#160; &#160;       Spacecraft angular position for
the bang-bang actuator<br>
     <b>thetaDotBang</b>   &#160;&#160;&#160; &#160;&#160;&#160; Spacecraft angular velocity for the bang-bang 
actuator<br>
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